50 VECTOR ANALYSIS. 



Whenever, therefore, o> is discontinuous at surfaces, the expressions 

 Pot V. to and NewV.w must be regarded as implicitly including the 

 surface-integrals 



cr' and 



IP ~ 



respectively, relating to such surfaces, and the expressions Pot V x ft> 

 and Lap V x ft) as including the surface-integrals 



/TV - c&r'xAo/ and 



L/>-*>Jo 



respectively, relating to such surfaces. 



101. We have already seen that if w is the] curl of any vector 

 function of position, V.o> = 0. (No. 68.) The converse is evidently 

 true, whenever the equation V.w = holds throughout all space, and 

 to has in general a definite potential ; for then 



ft) = VxT Lap ft>. 



4-7T 



Again, if V.o> = within any aperiphractic space A, contained 

 within finite boundaries, we may suppose that space to be enclosed by 

 a shell B having its inner surface coincident with the surface of A. 

 We may imagine a function of position ', such that ' = w in A, o>' = 

 outside of the shell B, and the integral jjjto.u) dv for B has the least 

 value consistent with the conditions that the normal component of w 

 at the outer surface is zero, and at the inner surface is equal to that 

 of ft), and that in the shell V.a/ = (compare No. 90). Then V.&/ = 

 throughout all space, and the potential of a/ will have in general a 

 definite value. Hence, 



ft/ = Vx i Lap ft>', 



47T 



and w will have the same value within the space A. 



|102. Def. If a) is a vector function of position in space, the Max- 

 wdlian * of co is a scalar function of position defined by the equation 



(Compare No. 92.) From this definition the following properties are 

 easily derived. It is supposed that the functions w and u are such 

 that their potentials have in general definite values. 



Max w = V. Pot w = Pot V. w, 

 V Max w = VV. Pot w = New V. , 

 Max Vu = 4f7ru, 



4-Trft) = V xLap w V Max ft). 



* The frequent occurrence of the integral in Maxwell's Treatise on Electricity and 

 Magnetism has suggested this name. 



t[The foregoing portion of this paper was printed in 1881, the rest in 1884.] 



